Structural properties and optical response of Na clusters in Ne, Ar, and Kr matrices
aa r X i v : . [ c ond - m a t . o t h e r] J un Structural properties and optical response of Na clusters in Ne, Ar, and Kr matrices
F. Fehrer
Institut f¨ur Theoretische Physik, Universit¨at Erlangen, Staudtstrasse 7, D-91058 Erlangen, Germany
P. M. Dinh and E. Suraud
Laboratoire de Physique Th´eorique, IRSAMC, CNRS, Universit´e de Toulouse,118 route de Narbonne F-31062 Toulouse C´edex, France
P.-G. Reinhard
Institut f¨ur Theoretische Physik, Universit¨at Erlangen,Staudtstrasse 7, D-91058 Erlangen, Germany andLaboratoire de Physique Th´eorique, IRSAMC, CNRS, Universit´e de Toulouse,118 route de Narbonne F-31062 Toulouse C´edex, France
We discuss the structural properties and optical response of a small Na cluster inside rare gas(RG) matrices of Ne, Ar, or Kr atoms. The mixed systems are described with a hierarchical model,treating the cluster at a quantum mechanical level and the matrix atoms classically in terms oftheir positions and polarizations. We pay special attention to the differences caused by the differentmatrix types. These differences can be explained by the interplay of core repulsion and dipoleattraction in the interaction between the cluster electrons and the RG atoms.
PACS numbers: 36.40.Gk,36.40.Jn,36.40.Qv,36.40.Sx
I. INTRODUCTION
The study of the properties of clusters embedded in amatrix or deposited on a substrate has motivated manyresearches since several years . This setup becomes in-creasingly important because the embedding/depositingsimplifies the experimental handling and because it is ageneric test case for composite materials. As in free clus-ters, the doorway to (laser induced) cluster dynamics isthe optical response, especially in metal clusters wherethe Mie plasmon dominates the optical properties . Letus mention as examples the systematics of optical re-sponse in large noble-metal clusters and its depen-dence on the environment . The study of the opticalresponse thus constitutes the key for understanding theresponse of clusters to electromagnetic probes and it alsoserves as a powerful tool for analyzing the underlyingcluster structure.The case of inert environments is especially interestingbecause it implies only moderate perturbations of clusterproperties. One can thus benefit from the well definedconditions from the surrounding system and still accesspredominantly the cluster properties. But the theoreticalmodeling becomes much more involved and the develop-ment of reliable as well as inexpensive approaches is stilla timely task, in particular what truly dynamical appli-cations are concerned . Nonetheless, fully detailedcalculations have been undertaken where details count,e.g. for the structure of small Na clusters on NaCl or the deposit dynamics of Pd clusters on a MgO sub-strate . But the expense for a fully fledged quantumsimulation grows huge. These subtle models are hardlyextendable to truly dynamical situations, to larger clus-ters or substrates, and to systematic explorations forbroad variations of conditions. This holds true not only for clusters on substrates, but for all composite systems.Thus there exists a great manifold of approximationswhich aim at an affordable compromise between reliabil-ity and expense. One route, for example, keeps all con-stituents at the same level, but simplifies the descriptionin terms of a microscopically founded tight-binding ap-proach . At the other extreme, one can con-sider all degrees of freedom as classical and perform puremolecular dynamics, as e.g. the deposition dynamics ofCu clusters on metal or Ar surface, and of Al or Auclusters on SiO . In between, one can take advantageof the very different importance or activity within thecomposite and thus develop a hierarchical modeling us-ing various levels of approximation for the different sub-systems. Such approaches are widely used in quantumchemistry, often called quantum-mechanical-molecular-mechanical (QM/MM) model. They have been appliedfor instance to chromophores in bio-molecules , sur-face physics , materials physics , embeddedmolecules and ion channels of cell membranes . Avariant of mixed modeling was applied in the case of aCs atom in He environment, the latter also considered asa quantum system .We are dealing here with Na clusters in rare-gas en-vironments (Ne, Ar, Kr). The large difference betweencluster metals (reactive) and rare gas (inert) naturallysuggests a hierarchical model where the substrate atomsare handled at a lower level of description, as classicalparticles but with a dynamical polarizability. Taking upprevious developments from , we have developed, inthe spirit of QM/MM approaches, a hierarchical modelfor Na clusters in contact with Ar and applied it tostructure, optical response and to non-linear dynam-ics of embedded clusters . It is the aim of this paperto present a generalization to other types of rare gasesTypeset by REVTEX(RG), namely Kr and Ne, and a first comparative studyof the effects of different environments on structure andoptical response. As test cases, we consider a Na clusterembedded in RG clusters of various sizes. Strictly speak-ing, they are mixed clusters and properties which dependon the size of the RG system are specific to mixed clus-ters. But we use the mixed systems mainly as model forNa clusters embedded in a matrix and we thus considerrather large systems. Henceforth, we will use the notion“matrix” for the RG surroundings. II. MODEL
The model has been introduced and presented in detailin . However for sake of completeness, we recall in thissection the ingredients and a few relevant formulae.The degrees of freedom of the model are the wave-functions of valence electrons of the metal cluster, { ϕ n ( r ) , n = 1 ...N el } , the coordinates of the cluster’sNa + ion cores, { R I , I = 1 ...N ion } , of the Ar atomscores Ar Q + ), { R a , a = 1 ...N Ar } , and of the Ar va-lence clouds, { R ′ a , a = 1 ...N Ar } . From the given totalenergy, the corresponding equations of motion are de-rived in a standard manner by variation. This leads tothe (time-dependent) Kohn-Sham equations for the onesingle-particle wavefunctions ϕ n ( r ) of the cluster elec-trons, and Hamiltonian equations of motion for the otherthree degrees of freedom, thus treated by classical mole-cular dynamics (MD). For the valence cluster electrons,we use a density functional theory at the level of thetime-dependent local-density approximation (TDLDA),augmented with an average-density self-interaction cor-rection (ADSIC) . The density of these electrons isgiven naturally as defined in mean-field theories andreads ρ el ( r ) = P n | ϕ n ( r ) | . A RG atom is described by two constituents with opposite charge, positive RG coreand negative RG valence cloud, which allows a correctdescription of polarization dynamics. In order to avoidsingularities, we associate a smooth (Gaussian) chargedistribution to both constituents having width σ RG ofthe order of the p shell ”size” in RG atoms, in the spiritof : ρ RG ,a ( r ) = e Qπ / σ ×× h exp (cid:18) − ( r − R a ) σ (cid:19) − exp (cid:18) − ( r − R ′ a ) σ (cid:19)i . (1)The corresponding Coulomb potential exerted by the RGatoms is related to the charge distribution (1) by thePoisson equation, and reads: V (pol)RG ,a ( r ) = e Q h erf ( | r − R a | /σ RG ) | r − R a |− erf ( | r − R ′ a | /σ RG ) | r − R ′ a | i , (2)where erf( r ) = √ π R r d x e − x stands for the error func-tion. As for the Na + ions, their dynamical polarizabilityis neglected and we treat them simply as charged pointparticles.The total energy of the system is composed as: E total = E Nacluster + E RG + E coupl + E VdW . (3)The energy of the Na cluster E Nacluster consists out ofTDLDA (with SIC) for the electrons, MD for ions, anda coupling of both by soft, local pseudo-potentials, fordetails see . The RG system and its coupling tothe cluster are described by E RG = X a P a M RG + X a P ′ a m RG + 12 k RG ( R ′ a − R a ) + X a
21 + e α Na /R (cid:18) α RG R + C Na , R + C Na , R (cid:19) + e α RG R R · ∇ R erf( R/ √ σ RG ) R . (8)The parameters are taken from literature for Na-Ne ,Na-Ar and Na-Kr . The pseudo-potential W el , RG in Eq. (5) for the electron-RG core repulsion has beenmodeled according to the proposal of : W el , RG ( r ) = e A el e β el ( r − r el ) , (9)with a final slight adjustment to the properties of a Na-RG molecule (bond length, binding energy, and opticalexcitation spectrum). Values of atomic and dimer prop-erties used are reported in table I. RG RG-Atom Na-RG RG bulk α RG [ a − ] IP [Ry] d [ a ] E [mRy] r s [ a ] E coh [mRy]Ne 2.67 1.585 10.01 0.0746 5.915 -0.0272Ar 11.08 1.158 9.47 0.3793 7.086 -0.1088Kr 16.79 1.029 9.29 0.6238 7.540 -0.1497TABLE I: Properties of the RG atoms and Na-RG dimersfor Ne, Ar, and Kr which were used for the fine-tuning of themodel. III. MODUS OPERANDI The numerical solution proceeds with standard meth-ods as described in detail in . The TDLDA equations for the cluster electrons are solved on a grid in coordi-nate space, using a time-splitting method for the propa-gation and accelerated gradient iterations for the station-ary solution. We furthermore employ the cylindrically-averaged pseudo-potential scheme (CAPS) as an approx-imation for the electrons , which is justified for thechosen embedded Na cluster. We have checked that a2D calculation with CAPS and a full 3D treatment of thevalence electron wavefunctions both give almost identicaloptical responses. The Na + ions as well as the RG atomsare treated in full 3D. The dynamics of the Na electronsis coupled to the response of the RG dipoles. However,the ionic and atomic positions can safely be frozen forthe present study as we focus on exploring the opticalresponse of the embedded clusters.To find an optimal Na+RG configuration, one startswith a fcc RG crystal, cuts from that a given number ofclosed shells, and cools the resulting configuration for apure RG cluster. One then carves a cavity of 13 atoms(Ar, Kr) or 19 atoms (Ne) from the center and placesthe Na cluster into it. This mixed configuration is re-optimized by means of successively cooled molecular dy-namics for the ions and atoms coupled to the stationarysolution for the cluster electrons.The stationary solution of the equations of motion pro-vides the ground state of the mixed system and consti-tutes the initial condition for further dynamical calcu-lations. One can then compute several observables toanalyze both the statics (structural properties) and thedynamics. A global measure for ionic and electronic clus-ter structure are the r.m.s. radii, r I,e = q h ( x + y + z ) i I,e (10)where h . . . i I = P I . . . and h . . . i e = R d r ρ e ( r ) . . . Notethat these quantities may also be used for characteriz-ing dynamics, although they take interest mostly on verylong times when ions and atoms actually move. This lat-ter aspect is not directly addressed here and we shall thususe them only as static quantities. We also evaluate theinsertion energy E ins which is defined as E ins = E tot (Na RG N ) + E (RG p ) − E (RG N+p ) − E (Na ) , (11)with p = 13 in the case of Ar and Kr, and p = 19 forNe. A further useful energetic observable is the ionizationpotential (IP), that is the energy required to remove oneelectron from the metal cluster. We compute it as thesingle-particle energy of the least bound electron whichis a reliable measure in ADSIC .At the side of truly dynamical properties, as alreadyemphasized, one should remind the especially importantrole played by the optical response. This observable iscomputed in an explicitely dynamical way. The dynamicsis initiated by an instantaneous dipole boost of the clus-ter electrons. The optical response is then obtained byspectral analysis of the emerging time-dependent dipolesignal following the strategy proposed in . IV. RESULTS AND DISCUSSION Figure 1 shows the RG structure in terms of radial dis-tributions of atoms. The effect of embedding is visualizedby comparing the pure RG system (dotted) with the RGdistribution around the Na cluster (full lines). The dis-tributions line up nicely in radial shells. For pure RGclusters, they remain very close to the radial shells of thebulk fcc structure (not shown here, for the case of Ar,see ). The overall scale is basically given by the bulkWigner-Seitz radius r s given in table I. Note that Nehas a much smaller r s resulting in denser packing as seenin figure 1. Carving of the cavity and insertion of Na has only small effect in the Ar and Kr environment andmainly for a few inner shells. For Ne, however, we see astronger perturbation which spreads over all atoms. Theexamples in figure 1 concern rather larger RG systemsprobably close to bulk. The embedding effects increasewith decreasing number of RG shells, remaining smallthroughout for Kr and Ar, but soon destroying any clearshell structure for Ne . The reason is that Ne is muchless bound than Ar or Kr as can be read of from thecohesion energy in table I, and already these seeminglymore robust materials are weakly bound.The impact of the RG environment on metal clusterproperties is analyzed in figure 2, which shows global ob-servables of the ground state configurations of Na inthe various matrices, as a function of matrix size. Theinsertion energies in the uppermost panel indicate thatNe cannot finally capture Na inside while Ar, and evenmore so Kr, provide robust environments for embeddedmetal clusters. The binding increases slowly with matrixsize, except for Ne where the polarizability is too weak toaccumulate sufficient long range attraction. The IP (sec-ond row in figure 2) makes a jump down from free Na tothe embedded cases and then stabilizes with a few fluc-tuations for small matrices and a faint further decreasefor larger ones. The energies are at the same scale forall RG types. The sudden drop from free to embeddedfor Ar and Kr is due to the core repulsion from the firstRG shell exerted on the cluster electrons. Adding furthershells acts only indirectly by compressing the whole ma-trix and thus bringing the innermost shell slightly closerto the cluster. Radii in Na are shown in the two lowerrows. They vary very little in general. There remaininteresting differences in detail. The trends with matrixsize are the same for electrons and ions. However thestep from free to embedded is much different to the ex-tent that electrons are more compressed when embeddedwhich is a visible effect from core repulsion, similar as thejump in IP. The radii decrease slightly with matrix sizefor Ne and increase for Ar or Kr. This indicates that corerepulsion prevails in Ne while dipole attraction becomesmore effective in Ar and Kr.We finally present in figure 3 the trends of the Na plas-mon resonance peak. The changes are generally small,at an absolute scale at limits of our modeling (whichwe estimate to about 0.1 eV uncertain). The relative trends, however, can be taken at smaller energy scaleand these carry several interesting aspects. As previ-ously discussed , the position of the peak results froma subtle cancellation between core repulsion and polar-ization effects. The step from free to embedded clustersfirst produces a blue shift because the cluster electronsfeel the core repulsion from the first layer of RG atoms.The plasmon peak moves slowly back to red with increas-ing system size because each new RG shell adds to thelong-range and attractive polarization potential. Polar-izability is also the key to the trend with RG material.It increases with atomic number (see column 2 of tableI). The cancellation mentioned above is thus more effec-tive in Kr than in Ar, and in Ar more than in Ne. Thisexplains the steady decrease of the peak position fromNe to Kr and the growth of slope with atomic number.In fact, Ne shows no significant slope to red at all. Thishappens because added atoms compress slightly the in-nermost shell (see shrinking radii with increasing systemsize in figure 2) and enhance core repulsion which, inturn, compensates the growth of polarization effects. Arand Kr experience no such compression and have anywaythe stronger polarizability.Experimental data for Na clusters embedded in raregas material are not yet available. A direct compari-son has thus to be postponed. But there exist alreadysome data on the optical response in the somehow sim-ilar combination of Ag clusters in RG material: theLausanne group has studied, among others, Ag @Ar ,Ag @Ar,Kr,Xe , and Ag @Ar,Ne in large RG matri-ces and, the Rostock group Ag covered by small layersof Ne, Ar, Kr and Xe (of sizes between 4 and 135 RGatoms) all immersed in a He droplet .Table II summarizes the results in terms of shifts ofthe plasmon peak position with changing RG material.A direct comparison with free neutral Ag clusters is notavailable because these are hard to handle experimen-tally. As for experiments performed in , Ag is embed-ded in very small RG matrices, and the whole systemitself embedded in a He droplet, for reasons of betterhandling. It is usually claimed that the helium envi-ronment interacts very faintly with the embedded sys-tem and can then be considered practically as a vacuum.However experimental data and DFT calculations show that the presence of He around Cs atoms producesa blue shift of the plasmon peak. It is thus likely thatsome small blue shift exists also for Ag clusters directlyin He droplets. Therefore a comparison with truly freeclusters is excluded. On the contrary, the influence ofHe around the RG layers in the experiments of is mostprobably negligible. The effect of the He droplet is in-deed strongly shielded by the RG layers both spatiallyand energetically. Remind that a typical RG-He bondhas a length of about 6–7 a and an energy of a fewmeV , comparable to the metal-RG bonds (seefourth column of table I). But once coated with the RGlayers, the metal cluster lies typically twice farther awayfrom the He droplet than in the case without RG layers, 10 13 16 19 22 25 28 31distance (a ) Ne r a d i a l d i s t r i bu t i on f un c t i on ArKr FIG. 1: Radial distribution of RG atoms in Na Ne , Na Ar , and Na Kr (full lines) compared with the distributions forthe pure RG cluster with 447 atoms before carving the cavity (dashed lines). -0.04-0.02 0 0.02 ene r g y i n s ( R y ) Ne Ar Kr I P ( R y ) r m s e l ( a ) r m s N a ( a ) nr. of RG-atoms FIG. 2: Ground state observables (energies, radii) for Na embedded in Ne, Ar, and Kr matrices of different size. Ag @ (from ) Ag @ (from ) Ag @ (from ) Ag @ (from )He drop → Ne : 0.009Ne → Ar: − . drop → Ar: − . 03 He drop → Ar : − . → Kr: − . / − . 09 He drop → Kr : − . drop → Xe : − . ω M i e ( e V ) nr. RG-atomsNeArKr FIG. 3: Plasmon resonance energies for Na in matrices ofdifferent size and RG material. whence a vanishingly small residual interaction betweenthe metal cluster and the He droplet. Thus the rela-tive shifts that we observe fby changing RG material aremost probably reliable. Finally, a word of caution is inorder. The above mentioned experimental measurementsand our calculations on embedded Na clusters reporta broadened, often even fragmented, peak with a widthsomewhat larger than the shifts we are looking at. Thecomparison is thus at the edge of experimental and the-oretical resolution.The data in table II agree with our theoretical resultsin that all shifts are very small. Inert environment turnsout to be indeed inert with respect to the plasmon peakposition. Looking in more detail at the relative shifts,one can read for the step from Ne to Ar a red shift of0.1 eV for Ag in large matrices and of 0.02 eV forAg in small matrices . The step from Ar to Kr yields0.06–0.09 eV for Ag in large systems and 0.03 eV forAg in small systems . The experiments with small RGlayers in a He droplet thus yield generally smaller redshifts. This also holds for the step from pure He to Arenvironment. This is probably explained by the size ofthe matrix, as we see from our results in figure 3 a slowbut steady move towards red (however, systematic er-rors by comparing two very different experiments cannotsafely be excluded). Our results for embedded Na clus-ters shown in figure 3 show a red shift of 0.1–0.2 eV forthe step from Ne to Ar and of 0.15–0.2 eV for Ar to Kr,both growing with increasing system size. They confirmall trends seen in experiment but are generally half anorder of magnitude larger. This is probably due to thesmaller Wigner-Seitz radius of Ag (3 a instead of 4 a for Na) which means that Ag structures are much morecompact than Na ones and thus couple less strongly tothe matrix because both fill the same RG cavity. Andthis, in turn, produces smaller shifts. In order to checkthat argument, we have simulated an embedded “Ag ”cluster simply by rescaling the ionic positions of the Na by the ratio of Wigner-Seitz radii, that is 3/4, and byreoptimizing the RG positions before the calculations ofthe optical response of this pseudo Ag . We find thesame trends as with Na in figure 3, but indeed reducedby a factor of three.Besides, previous theoretical calculations were per-formed within TDLDA and a jellium model, coupling tothe RG materials in terms of a static dielectric mediumwith dielectric constant ε for different metal clusters,namely K , Na and Al , and Ag . They always yieldred shifts which, moreover, increase with increasing ε .Indeed, the ingredients added in these models can onlyproduce long-range polarization effects and thus can onlygive a red shift. Our model contains as new component,the RG short-range and repulsive core potentials whichgenerate a blue shift of the plasmon peak. A clear exper-imental assessment would yet require a comparison withtruly free neutral metal clusters. V. CONCLUSIONS We have discussed in this paper static properties andoptical response of a small Na cluster embedded in var-ious rare gas (RG) matrices. For this purpose, we useda recently introduced hierarchical approach combininga fully detailed quantum-mechanical description of thecluster with a classical modeling for the RG environmentand its interactions with the cluster. We have studiedeffects from embedding in RG environment with up to434 atoms with particular emphasis on the change withRG material. Various observables were considered. Theinsertion energy yields stable embedding for Ar and Krbut not for Ne within the considered system sizes. TheIP behaves the same in all three materials: It drops fromfree to embedded and stays nearly constant for all ma-trix sizes. The electronic and ionic radii of the Na clusterchange very little; the most noteworthy effect is here aslight compression of the electron cloud through embed-ding. What optical response is concerned, we studied theeffect of the RG matrices on the position of the surfaceplasmon peak in the metal cluster. The net shift fromfree to embedded clusters is very small due to a nearcancellation of the blue shift from core repulsion withthe red shift from dipole polarization in the interactionwith the RG atoms. The polarization increases with theRG atomic weight and thus final peak position goes fromblue shift to red shift on the way from Ne over Ar toKr. The polarization effect also increases with the size ofthe RG matrix which produces a slow and steady trendto red with increasing system size. Comparison with ex-perimental data on embedded Ag clusters confirms thesetrends and orders of magnitude. 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